Once while solving problems from Romanian Mathematical Olympiads I encountered a mathematical gem. The problem I found was asking to calcualte the following integral
What do I find especially beautiful about this problem? It rewards the desire to examine a bit more than is asked initially!
Solution. Let's try to evaluate one more integral
Can you guess the next step?
Yes, we will take advantage of the fact that the sum of integrals is the integral of sum. In other words:
On other hand we can easily calculate integrals of the form . The derivative of is equal to , which can also be expressed as a linear combination of our two integrals, i.e.
Now we simply have to solve a system of linear equations. We obtain and
Has anyone of you seen problems with the similar ideas? How often while solving a problem you investigate possible variations of the conditions?